In the context of
disasters, a firestorm is a fire large and hot enough to cause a local low pressure area, which in turn pulls
oxygenated air towards the fire's base. Enough will flow in to make it not only self-sustaining on the most minimal
fuel but also able to reach extremely high temperatures. The fire will spread typically through tossing large quantities of burning material or ash up the 'furnace' which then settles on unburnt areas nearby.
Firestorms can be the deadliest part of aerial bombardment. In Dresden, Germany during World War II, the Allies unleashed a hellish amount of incendiary bombs into the center of the city. Rationales for targeting it vary; although it was a railroad hub, it was also mostly beautiful historic buildings; however, there is reason to believe the Allies may not have known precisely what was going to take place. This, naturally, does nothing to lessen the enormity of the act.
In any case, after the mass incendiary attack, a firestorm started. Reports from survivors indicate that at the edges of the storm, there was a flow of air into the center such that at ground level a 200-mph wind was blowing everything and everyone in its path into the fire. The heat reached such intense levels that people who had sought shelters belowground, down as far as a hundred feet, were baked in their supposed refuges and left as blackened charcoal. For an excellent description, see the Dresden node.
This incident figures somewhat prominently in the Kurt Vonnegut book Slaughterhouse Five.
Note that this is also the most likely result of a nuclear airburst over many of the world's cities. The flash from a nuclear weapon is enough to ignite most anything at close to medium range. If it does so all at once (as it would) this, coupled with the enormous updraft and dynamic overpressure of air rushing in to fill the space left by atmosphere turned to plasma and sent soaring upward in the characteristic mushroom cloud would likely produce a massive firestorm nearly instantly.
The U.S. military, however, somehow never figured this into its tally of predicted 'Effects of a Nuclear Strike.' This is one reason that targeting during the Cold War climbed to such enormous numbers of warheads dedicated to destroy relatively soft targets. Typically, the Air Force would only calculate blast effects. If pushed, they might consider prompt radiation and flash damage. Firestorms, however, weren't discussed. Given that, using ten warheads in a grid over a city suddenly seems really conservative instead of downright ludicrous.
Besides, it let them balloon their nuclear budgets.